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The finite product law has iid coordinate maps

Statement

Let $A$ be finite and nonempty, let $p$ be a probability vector on $A$ , and let $n \geq 1$ . The finite mass assignment $p^{\otimes n}$ of Finite product mass on words is a probability law on $A^{n}$ , and its coordinate maps $X_{i} (x^{n}) = x_{i}$ are independent random elements, each with law $p$ .

Facts & Assumptions

Given: A finite nonempty alphabet $A$ , a probability vector $p$ on $A$ , and $n \geq 1$ .

[F1]

Finite product mass on words: $p^{\otimes n} ({x^{n}}) = \prod_{i = 1}^{n} p (x_{i})$ , event masses are sums of these atom weights, and $X_{i} (x^{n}) = x_{i}$ on the full power-set $σ$ -algebra.

[F2]

Finite distributions, entropy, divergence, and stochastic matrices: a probability vector has nonnegative coordinates summing to one.

[F3]

Finite probability spaces, outcome weights, events, and event probabilities: a finite probability space is a finite outcome set with nonnegative weights summing to one, and event probabilities are the corresponding finite sums.

[F4]

A measurable function between measurable spaces: a map is measurable when the preimage of each measurable target set is measurable in the domain.

[F5]

Random elements and real random variables: a random element is a measurable map from a probability space to a measurable space.

[F6]

The sigma-algebra generated by a function: $σ (X_{i}) = {X_{i}^{- 1} (B) : B \in B (A)}$ ; for the finite discrete alphabet, every subset is Borel.

[F7]

Independent sigma-algebras and independent events: sigma-algebras are independent when their event classes are independent as classes of events.

[F8]

Independent families of event classes: independence of event classes means the probability of every finite intersection, choosing one event from each selected class, equals the product of those probabilities.

[F9]

Independent random elements: random elements are independent when their generated sigma-algebras are independent.

Proof

1.1F1F2F3algebra

By [F1] and finite distributivity, $\sum_{x^{n} \in A^{n}} p^{\otimes n} ({x^{n}}) = \prod_{i = 1}^{n} \sum_{a \in A} p (a) = 1$ by [F2], and all atom weights are nonnegative. Thus [F3] makes the assignment $p^{\otimes n}$ a finite probability law with the event masses defined in [F1].

2.1F1F2F4F5F6F7F8F9step 1.1algebra∎

Each $X_{i}$ is measurable by [F4], because every target subset has preimage in the full power set of $Ω_{n}$ ; it is therefore a random element by [F5]. For arbitrary subsets $B_{i} \subseteq A$ , put $E_{i} = X_{i}^{- 1} (B_{i}) \in σ (X_{i})$ by [F6]. The product-mass formula [F1] and finite distributivity give $p^{\otimes n} (⋂_{i = 1}^{n} E_{i}) = \sum_{x^{n} \in B_{1} \times \dots \times B_{n}} \prod_{i = 1}^{n} p (x_{i}) = \prod_{i = 1}^{n} \sum_{a \in B_{i}} p (a)$ . By step 1.1 and [F2], each factor is $p^{\otimes n} (E_{i})$ after summing the other coordinates over $A$ . Every event in each $σ (X_{i})$ has this form by [F6]; setting omitted coordinates to $A$ gives the same factorization for every finite subfamily. Thus [F7] and [F8] make the coordinate sigma-algebras independent, and [F9] makes the coordinate random elements independent. Taking $B_{j} = A$ for $j \neq i$ shows $X_{i}$ has law $p$ .

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Sources

Thomas M. Cover and Joy A. Thomas, Elements of Information Theory, second edition, Chapter 11 (standard reference, not scraped)
Yury Polyanskiy and Yihong Wu, Information Theory: From Coding to Learning, author-hosted book draft (standard reference, not scraped)